↑

Properties

Label 2-1575-1.1-c3-0-100
Degree 22
Conductor 15751575
Sign −1-1
Analytic cond. 92.928092.9280
Root an. cond. 9.639919.63991
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 0.561·2-s − 7.68·4-s − 7·7-s − 8.80·8-s + 25.8·11-s − 15.5·13-s − 3.93·14-s + 56.5·16-s + 95.1·17-s − 143.·19-s + 14.4·22-s − 77.5·23-s − 8.73·26-s + 53.7·28-s + 204.·29-s − 40.6·31-s + 102.·32-s + 53.4·34-s + 95.6·37-s − 80.5·38-s − 24.7·41-s + 282.·43-s − 198.·44-s − 43.5·46-s − 257.·47-s + 49·49-s + 119.·52-s + ⋯
L(s)  = 1  + 0.198·2-s − 0.960·4-s − 0.377·7-s − 0.389·8-s + 0.707·11-s − 0.331·13-s − 0.0750·14-s + 0.883·16-s + 1.35·17-s − 1.73·19-s + 0.140·22-s − 0.702·23-s − 0.0658·26-s + 0.363·28-s + 1.30·29-s − 0.235·31-s + 0.564·32-s + 0.269·34-s + 0.425·37-s − 0.343·38-s − 0.0941·41-s + 1.00·43-s − 0.679·44-s − 0.139·46-s − 0.798·47-s + 0.142·49-s + 0.318·52-s + ⋯

Functional equation

Λ(s)=(1575s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(1575s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 15751575    =    32⋅52⋅73^{2} \cdot 5^{2} \cdot 7
Sign: −1-1
Analytic conductor: 92.928092.9280
Root analytic conductor: 9.639919.63991
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1575, ( :3/2), −1)(2,\ 1575,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1 1
5 1 1
7 1+7T 1 + 7T
good2 1−0.561T+8T2 1 - 0.561T + 8T^{2}
11 1−25.8T+1.33e3T2 1 - 25.8T + 1.33e3T^{2}
13 1+15.5T+2.19e3T2 1 + 15.5T + 2.19e3T^{2}
17 1−95.1T+4.91e3T2 1 - 95.1T + 4.91e3T^{2}
19 1+143.T+6.85e3T2 1 + 143.T + 6.85e3T^{2}
23 1+77.5T+1.21e4T2 1 + 77.5T + 1.21e4T^{2}
29 1−204.T+2.43e4T2 1 - 204.T + 2.43e4T^{2}
31 1+40.6T+2.97e4T2 1 + 40.6T + 2.97e4T^{2}
37 1−95.6T+5.06e4T2 1 - 95.6T + 5.06e4T^{2}
41 1+24.7T+6.89e4T2 1 + 24.7T + 6.89e4T^{2}
43 1−282.T+7.95e4T2 1 - 282.T + 7.95e4T^{2}
47 1+257.T+1.03e5T2 1 + 257.T + 1.03e5T^{2}
53 1+257.T+1.48e5T2 1 + 257.T + 1.48e5T^{2}
59 1−651.T+2.05e5T2 1 - 651.T + 2.05e5T^{2}
61 1−451.T+2.26e5T2 1 - 451.T + 2.26e5T^{2}
67 1−832.T+3.00e5T2 1 - 832.T + 3.00e5T^{2}
71 1−174.T+3.57e5T2 1 - 174.T + 3.57e5T^{2}
73 1+47.4T+3.89e5T2 1 + 47.4T + 3.89e5T^{2}
79 1+1.16T+4.93e5T2 1 + 1.16T + 4.93e5T^{2}
83 1+1.49e3T+5.71e5T2 1 + 1.49e3T + 5.71e5T^{2}
89 1+1.30e3T+7.04e5T2 1 + 1.30e3T + 7.04e5T^{2}
97 1+1.36e3T+9.12e5T2 1 + 1.36e3T + 9.12e5T^{2}
show more
show less
   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−8.522159381079869159984886151380, −8.170915117258542673441349737224, −6.94308381369935715011788674764, −6.13648521869361046893741555063, −5.34151541851832069222826536003, −4.31721906385571152202760831783, −3.75219163088503200248260963520, −2.61079507159791105883210867362, −1.15656329597800068966987410115, 0, 1.15656329597800068966987410115, 2.61079507159791105883210867362, 3.75219163088503200248260963520, 4.31721906385571152202760831783, 5.34151541851832069222826536003, 6.13648521869361046893741555063, 6.94308381369935715011788674764, 8.170915117258542673441349737224, 8.522159381079869159984886151380

Graph of the ZZ-function along the critical line